Subgroup ($H$) information
| Description: | $C_{12}.D_6$ |
| Order: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Index: | \(1100\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rrrr}
0 & 8 & 8 & 8 \\
8 & 2 & 10 & 0 \\
9 & 10 & 8 & 1 \\
5 & 3 & 4 & 0
\end{array}\right), \left(\begin{array}{rrrr}
5 & 4 & 5 & 6 \\
10 & 6 & 10 & 10 \\
4 & 9 & 9 & 0 \\
9 & 5 & 10 & 8
\end{array}\right), \left(\begin{array}{rrrr}
1 & 4 & 3 & 8 \\
6 & 4 & 3 & 4 \\
7 & 8 & 1 & 10 \\
9 & 5 & 9 & 8
\end{array}\right), \left(\begin{array}{rrrr}
8 & 7 & 2 & 8 \\
10 & 7 & 3 & 5 \\
9 & 8 & 4 & 2 \\
9 & 3 & 2 & 4
\end{array}\right), \left(\begin{array}{rrrr}
0 & 5 & 0 & 2 \\
9 & 6 & 8 & 0 \\
7 & 9 & 5 & 6 \\
10 & 7 & 2 & 0
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
0 & 10 & 0 & 0 \\
0 & 0 & 10 & 0 \\
0 & 0 & 0 & 10
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_{15}:Q_8\times \SL(2,11)$ |
| Order: | \(158400\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^5\times S_3).C_2^2.\PSL(2,11).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2\times \SL(2,3).C_2^6$, of order \(27648\)\(\medspace = 2^{10} \cdot 3^{3} \) |
| $W$ | $D_6^2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $55$ |
| Möbius function | not computed |
| Projective image | not computed |